Technology for a better society 1 Optimization-based train dispatching systems in operation in Europe INFORMS Annual Meeting San Francisco, November 11

Technology for a better society 1

Optimization-based train dispatching systems in operation in Europe

INFORMS Annual Meeting

San Francisco, November 11

Leonardo LamorgeseSINTEF (Oslo)Joint work with

Carlo Mannino, Arnt-Gunnar LiumSINTEF


Outline

• Introduction to Train Dispatching (TD)• Modelling and solving TD• Real-life implementations of dispatching systems

deploying optimization

Technology for a better society

• Train movements are controlled by human operators (dispatchers)

• Dispatchers control railway traffic by switches, traffic lights, phone calls etc.

• When trains deviate from the official timetable, dispatchers must take re-routing and re-scheduling decisions.

• The goal is to alleviate overall delays, knock on effects and to return to the official timetable as soon as possible.

3

Train dispatching

Kilde: Togleder.no


Capacity utilization at peak hour

4


• Each dispatching central is responsible for the train movements in a region.

• Each dispatcher is responsible for a line or parts of a line that is under the control of the given central.

• (Almost) total lack of decision support.

• All up to experience and a set of predetermined rules.

5

"Traditional" train dispatching

Is there a better way?Kilde: Togleder.no


The Train Dispatching (optimization!) problem

Given:

•set of networks N = {(T1, R1), …, (Tn, Rn)} •fading matrix [Atj]tT, jR

•frequency domain Ft, tR•revenue function u(C(p,f, s)) = u(p, f, s) of the coverage

Givena railway network with its current and (near-) future status, a set of trains with their current positions, expected speeds and a timetable

Find in real time: a route for every train and a conflict-free schedule minimizing (a measure of) the deviation from the wanted timetable

The Train Dispatching problem

• Very hard combinatorial optimization problem (in theory and practice)• Job-shop scheduling problem with routing


Some Literature Papers: Acuna-Agost, Adenso-Diaz, Afonso, Aronsson, Ahuja, Bisbo, Bohlin,

Caimi, Cangalovic, Chiu, Chou, Conti, Corman, Cunha, D'Ariano, de Aquino, de Carvalho, Dollowet, Ehrgott Feillet, Ferreira, Feung, Flammini, Fukumura, Gatto, Gonzalez, Gonzalez-Torre, Gueye, Harrod, Higgings, Huisman, Jacob, Kozan, Kreuger, Lamorgese, Larsen, Llanos Quintero, Laube, Lee, Leung, Lusby, Lüthi, Mannino, Mascis, Mazzarello, Medeossi, Michelon, Mladenovic, Nash, Ottaviani, Pacciarelli, Peeters, Persson, Pranzo, Rodriguez, Romeiro de Jesus, Ryan, Sahin, Sato, Schachteebeck, Schmidt, Schöbel , Sundaravalli, Widmayer Ph.D. Thesis (with surveys): Corman 2010, Lüthi 2009, Caimi 2009, D’Ariano 2008, Conti 2006, Törnquist 2006, Flammini 2005, Oliveira 2001 ….

Too many to get into details

Certainly missing many others

Carlo

Maybe a bit paradoxally

Carlo

and probbly more, these are those I have bumped into


• A train runs a (ordered) sequence of atomic resources

• Atomic movement u = (i,r): occupation of rail resource r by train i

Modelling train movements

• tu time train i enters resource r (u = (i,r))

r1

r2 r3r4

r0

• The network (line) can be decomposed into atomic resources

• Atomic resources can be occupied by at most one train at a time

u = (i,r)i

r

tu


• Successive movements: uvuv ltt Simple precedence

r1

r2 r3r4

r0

The route and competing trains

• Distinct trains, incompatible or non-sharable resources:

Disjunctive precedence 𝑡𝑢− 𝑡𝑤≥ 𝑙𝑤𝑢𝑡 𝑧−𝑡𝑣≥ 𝑙𝑧𝑣

ij

A BC

u

v

tu

tv


The Train Dispatching problem

Mathematical representations => MILP

Time-indexed formulations (binary scheduling variables)

Big-M formulations (continuous scheduling variables, indicator variables)

The routing problem (basically in stations) can be included in the MILP

Disjunctive programs are difficult!

n

zuzuvwvw

uvuv

Rt

Auzwv)ltt)ltt

Fvultt

tc

)},(),,({ ((

),(

)(min

Disjunctive program


A Big-M formulation

min c(t1)

At1 0 ≤ b – My1 scheduling on the line

0 Dt2 ≤ d – My2 scheduling in stations … ≤ … routing in stationsy {0,1}n , t ℝm,

t1 and t2 share variables associated with arrivals and departures from stations

Problem: large instances, weak formulation

i

Station 1 Track 1 Station 2 Track 2

y1 and y2 binary indicator variables controlling disjunctions


Benders Reformulation

MASTER

SLAVE

min c(t1)

At1 ≤ b – My1 scheduling on the line

Dt2 ≤ d – My2 scheduling in stations y {0,1}n , t ℝm, min c(t1)

At1 ≤ b – My1 scheduling on the line

Ey1 ≤ f combinatorial Benders Cuts

y1 {0,1}q , t1 ℝg,


Master: the Line Dispatching problem

i

Station 1 Track 1 Station 2 Track 2

Substitute each station sub-route with a single node

The Line Dispatching problem

Find a schedule t minimizing c(t) so that trains only meet in stations or in

multiple track regions.

T2 T1S1 S2

Output: arrivals and departure times in stations


Slave: the Station Dispatching problem

The Station Dispatching problem (feasibility)

Given arrival and departure times for trains in a station.

Find routes and a conflict-free schedule, or prove problem not feasible

The slave decomposes into independent problems, one for each station


A common case: fixed route Station Dispatching

• Ass. Single (fixed) route to each platform

15

The (fixed routes) Station Dispatching problem

Given arrival and departure times for trains in a station.

Find a feasible assignment of platforms to each train or show none exists


Fixed route Station Dispatching: a colouring problem

16

Th. if every platform can accomodate every train Station Dispatching is easy

Proof: reduction from colouring of interval graphs

Th. if platforms and trains have multiple lengths Station Dispatching is NP-complete

Proof: reduction from µ-colouring of interval graphs (Bonomo et al., 2006)


Solving the Train Dispatching problem

Solve the current master

Solve the slave(s)

(t1 , y1)

(t2 , y2, …)

Feasible?

Add

(t1 , y1 , t2 , y2, …) optimal

Line dispatching

Station dispatching

• The master problem solved by row/column generation

NO


* dismissed in 2008 (due to entire system renewal)** scheduled

A classification of optimization-based dispatching systems

What Technique Where From

MASS TRANSIT

Terminal Stations Exact: branch&bound Milano 2007*

Multiple Lines Min-cost flow based heuristic Dehli 2015**

MAIN LINE

Regional Lines Heuristic Italy, Latvia

2011, 2014

Regional Lines Exact: Benders' Norway 2014

Large Stations Heuristic/Exact: MILP Italy 2014

In principle: the disjunctive formulation applies to all systems In practice: different approaches suitable for different cases


• The system was put in operation in Stavanger in February 2014.

• From Stavanger to Moi (Jærbanen), 123 km, 16 stations, single- and double-tracks.

• Up to 120 trains per day.

• Solutions are presented to dispatchers through a space-time diagram (Train Graph)

19

Stavanger-Moi (Norway): dispatching tool


Stavanger-Moi: the Train Graph


Suggestions for how each and every train should drive for the next few hours are

provided to the dispatchers.

21

Meta-scheme

Database containing

information about the

infrastructure

Server providing us with information about train

movements (in real time)

Optimization module


Stavanger-Moi (Norway)


• Between 3000 and 5000 calls to the algorithm per day (on average)

• Over 90% of the problems solved to optimality within 10 seconds

• More details in: "An exact decomposition approach to the train dispatching problem", Lamorgese, Mannino to appear on Operations Research

23

Some statistics


"Optimal" solutions better than "good" ones

Delay ranges (mins) PerformanceMethod [0,5) [5,10) [10,15) 15+ Time (s) # fails

Heuristic 85.2 % 1.7 % 1.3 % 11.8% 0.7 413

Exact 91.2 % 4.0 % 1.7 % 3.1 % 4.3 10

Tests run 29.1.2013

Benchmarking the algorithm with current solutions in Italy

"Triple" regional line, centered in Foligno, using our heuristics

A "natural" Key Performance Indicator: # of delayed trains

Comparisons on 4 delay classes for 130 trains in one day


End

Documents

Technology for a better society 1 Optimization-based train dispatching systems in operation in Europe INFORMS Annual Meeting San Francisco, November 11